Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling

We propose a way of including the next-to-leading (NLO) order spin-spin coupling in an effective-one-body (EOB) Hamiltonian. This work extends by S. Balmelli and P. Jetzer, [Phys. Rev. D 87, 124036 (2013)], which is restricted to the case of equatorial orbits and aligned spins, to general orbits with arbitrary spin orientations. This is done by applying appropriate canonical phase-space transformations to the NLO spin-spin Hamiltonian in Arnowitt-Deser-Misner (ADM) coordinates, and systematically adding effective quantities at NLO to all spin-squared terms appearing in the EOB Hamiltonian. As required by consistency, the introduced quantities reduce to zero in the test-mass limit. We expose the result both in a general gauge and in a gauge-fixed form. The last is chosen such as to minimize the number of new coefficients that have to be inserted into the effective spin squared. As a result, the 25 parameters that describe the ADM NLO spin-spin dynamics get condensed into only 12 EOB terms.

Figure 1

Reduced angular frequency (top) and binding energy (bottom) at the LSO as a function of the effective spin parameter ${\chi }_0$, in the case of equal masses and equal spins, for different EOB models.